Problem: Simplify the following expression: $y = \dfrac{-9x^2- 34x+8}{-9x + 2}$
Answer: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-9)}{(8)} &=& -72 \\ {a} + {b} &=& &=& {-34} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-72$ and add them together. Remember, since $-72$ is negative, one of the factors must be negative. The factors that add up to ${-34}$ will be your ${a}$ and ${b}$ When ${a}$ is ${2}$ and ${b}$ is ${-36}$ $ \begin{eqnarray} {ab} &=& ({2})({-36}) &=& -72 \\ {a} + {b} &=& {2} + {-36} &=& -34 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({-9}x^2 +{2}x) + ({-36}x +{8}) $ Factor out the common factors: $ x(-9x + 2) + 4(-9x + 2)$ Now factor out $(-9x + 2)$ $ (-9x + 2)(x + 4)$ The original expression can therefore be written: $ \dfrac{(-9x + 2)(x + 4)}{-9x + 2}$ We are dividing by $-9x + 2$ , so $-9x + 2 \neq 0$ Therefore, $x \neq \frac{2}{9}$ This leaves us with $x + 4; x \neq \frac{2}{9}$.